If I had told you their product, you would have known what the numbers were.

If I had told you instead the sum of their squares, you would also have known what the numbers were.

But if I had told you instead the sum of the numbers, you wouldn't have been able to tell what the numbers were.

Which were the numbers?

If duplicates are not allowed

Any 4tuple 1,p1,p2,p3 has an  unique product if p1...,p3 are primes

To qualify as puzzle's solution we need to check  the  SUM of squares

e.g. 1 2  5 7    SofSQ= 1+4+25+49=79  ===

  Now check all possible 4tuples of distinct primes not over 7:

1 2 3 5  ==not 79
1 2 3 7   ==not 79
1 2 5 7  79
1 3 5 7 ==not 79
2 3 5 7  ==  not 79

so
1 2 5 7      is a valid solution of  the  puzzle